The algebraic procedure described in Sec. 6.5 is convenient for obtaining a rough estimate of the number of stages needed in fractional extraction, but is seldom accurate enough for design purposes, because distribution coefficients usually change as concentrations change from stage to stage. This section describes a numerical procedure that is generally applicable whenever distribution coefficients are known and the phases leaving an extraction stage are in equilibrium.

The procedure will be illustrated by the zirconium-hafnium separation example treated in Sec. 6.5. The material-balance quantities used for the present illustration are listed in Table 4.8. The relative flow rates of solvent, scrub, and feed streams are those recommended by Нигё and Saint James [H4], as are the concentrations of HN03 and NaN03 in solvent, scrub, and feed streams. Changes in volume of the aqueous and organic streams within the scrubbing section and within the extracting section are to be neglected. The concentrations of total TBP in the organic stream is specified as 2.25 mol/liter, and the slight solubility of TBP in water is to be neglected. The concentration of zirconium and hafnium in the aqueous feed, the required zirconium recovery, and the required hafnium decontamination factor are the same as in the example of Sec. 6.5; the concentrations of Zr(N03)4 and Hf(N03 )4 in the raffinate and extract streams are thereby specified.

After the above variables have been set, the cascade is fully specified and only one set of values for the number of stages in extracting and scrubbing sections will perform the specified separation. All other extractable components will be distributed in a determinate manner between aqueous-residue and organic-extract streams.

In the present example, nitric acid is an extractable component, whose split between extract and raffinate streams cannot be specified in advance. However, in the calculational procedure to be described, it is necessary to start with specified concentrations of all

Table 4.8 Material balance for zirconium-hafnium separation by fractional extraction with TBP+ In Out Stream: Phase: Feed Aqueous Scrub Aqueous Solvent Organic Total Residue Aqueous Extract Organic Total Gram-moles/ liter: TBP NaN03 3.5 3.5 2.25 — 3.5 2.25 HN03 3.0 3.0 1.6 — 3.03* 1.576* Zr(N03)4 0.123 0.000 0.000 — 0.00123 0.0578 Hf(N03)4 0.00246 0.000 0.000 _ 0.00122 5.78 X 10’6 Liters 48 48 100 — 96 100 Gram-moles: TBP — — 225 225 — 225 225 NaN03 168 168 336 336 — 336 HN03 144 144 160 448 291 158 449 Zr(N03)4 5.90 0.000 0.000 5.90 0.118 5.78 5.90 Hf(N03)4) 0.118 0.000 0.000 0.118 0.117 0.006 0.118 *Basis: 100 liters of solvent. * These concentrations cannot be specified in advance and must be confirmed by calculation or experiment.

components in terminal streams from the cascade, so that provisional values must be assumed for nitric acid concentration in the extract streams, and the corresponding concentration of nitric acid in the raffinate is obtained from a material balance. If subsequent calculation fails to confirm the correctness of these provisional values, new values must be assumed for acid concentration in the extract stream and the calculation repeated. The particular values of exit acid concentrations listed in Table 4.8 were arrived at by several iterative calculations and give a consistent analytical solution to the separation problem.

To obtain distribution coefficients as a function of concentration, it will be assumed that equilibria are established in the three following reactions:

H><?) + N03 ~(aq) + TBP(o) — HN03 — TBP(o) KH = 0.145

Zr4+(aq) + 4N03′(aq) + 2TBP(o) — Zr(N03)„-2TBP(o) KZr = 0.0032

Hf*+(aq) + 4N03 ‘(aq) + 2TBP(o) — Hf(N03)„ -2TBP(o) KH{ = 0.00032

The value of 0.145 for the equilibrium constant of the nitric acid complex is an average value derived from the equilibrium data of Moore [М2], Alcock et al. [Al], and Gruverman [G7], The value of 0.0032 for the zirconium equilibrium is the average value derived from the equilibrium data in Table 4.4. The value of 0.00032 for the hafnium equilibrium is derived from the separation factor of 10 measured for zirconium-hafnium mixtures by Hure and Saint James [Н4]. Distribution coefficients are then given by the following equations.

HN03:	DH — — — 0.145yTBPxNO — *H	(4.85)
Zr(N03)4:	£>zr = ^ = 0.0032Otbp)2(*ncv)4 xZt	(4.86)

Hf(N03)4: D^f — — 0.00032(>’xbp)*C5Cno,")4 (4-87)

*Hf

where Утвр is the concentration of uncombined TBP in the organic phase and xNOj — is the total nitrate concentration in the aqueous phase. For the concentrations assumed for Table 4.8, Утвр is obtained from

Утвр = 2.25 — уп + 20 Zr + yHf)] mol/liter (4.88)

where 2.25 is the concentration of TBP in all forms in the organic phase. The aqueous nitrate concentration is obtained from

xNOj — = 3.5 + xH + 4(x& + *Hf) mol/liter (4.89)

where 3.5 is the constant concentration of sodium nitrate in the aqueous phase.

To find the required number of stages, a calculation is made of the concentrations of zirconium nitrate, hafnium nitrate, and nitric acid in the organic phase as a function of stage number in the scrubbing and extracting sections. A plot similar to Fig. 4.19 is then made of the concentration of zirconium versus the concentration of hafnium in each section. The point of intersection of the curves gives the concentrations of these components in the organic phase flowing between the two sections at the feed point, at which

Zr: УІг^г+1 = y*zt, N (4.90)

Hf: yfi tM+l =ygf„ (4.91)

The number of stages in each section (M and N) at which these concentrations are equal is the result of the first trial calculation.

A check is then made to determine if the nitric acid concentration in the organic phase leaving the extracting section equals that in the organic phase entering the scrubbing section. If

yn, N = Ун, м+і (4-92)

the provisional values assumed for nitric acid concentrations in residue and extract streams are correct. If these concentrations are not equal, new values must be assumed for nitric acid concen­trations in the residue and extract streams and the entire calculational procedure repeated.

Steps in the calculation of the concentrations of zirconium nitrate, hafnium nitrate, nitric acid, and uncombined TBP as a function of stage number n in the extracting section are given in Table 4.9.

In order to calculate equilibrium concentrations уf in the organic stream leaving stage 1, jtnOj -,i is calculated from Eq. (4.89), a trial value of Утвр. і is assumed, and Df andyf are obtained from Eqs. (4.85) through (4.87). The trial value of yfBPjl is checked by using the yf in Eq. (4.88), and when a trial value of yfBP, i that satisfies this condition is found, a consistent set of concentrations in the streams leaving the first extracting stage has been obtained.

The next step is to apply the material-balance equation (4.51) to calculate the concen­trations in the aqueous phase leaving stage 2. The calculation then proceeds up through the extracting section by a repetition of these steps.

Steps in the calculation of concentrations as a function of stage number m in the scrubbing section are given in Table 4.10. Die calculation is started with the specified concentrations y? in the organic extract stream leaving stage 1, obtained from Table 4.8. Because the concentrations yf have been chosen, the value of у? вр, і can be calculated from Eq. (4.88), and a value of x$0j-,i must be assumed to calculate values of Df and xf from Eqs. (4.85) through (4.87). Mien the assumed value of д$о,-,і agrees with that obtained by substituting the corresponding values of xf in Eq. (4.89), a consistent set of concentrations in the streams leaving the first scrubbing stage has been obtained.

Stage no. n	Aqueo 4 =	us concentratiot rf + E {yE	1, mol/liter — і ~Уо)	Total N03’a E	Assumed free TBP in organic phase E ТТВР. п	Distribution coefficients	Organic concentration, mol/liter yE ~ D4 -‘ft n
5 + F^
HN03 >H, n	Zr(N03)4 .КІг. п	Hf(N03)4 Ут, п	Free TBP* E УТВ?,п
hno3 rE xH, n	Zr<N03)4 xZr, n	Hf(N03)4 „£ xHf, n	HN03 Dnb	Zr(N03)4 DzrC	Hf(NOs)4 DHfd
і	3.03^	0.001230^	0.001224/	6.53	0.570	0.540	1.897	0.1897	1.635	0.00233	0.000232	0.610
					0.580	0.550	1.967	0.1967	1.664	0.00242	0.000241	0.580
2	3.09	0.00375	0.001475	6.61	0.565	0.542	1.954	0.1954	1.678	0.00732	0.000288	0.577
					0.563	0.540	1.944	0.1944	1.671	0.00729	0.000287	0.563
3	3.10	0.00882	0.001523	6.64	0.555	0.534	1.918	0.1918	1.657	0.01692	0.000292	0.558
					0.556	0.535	1.924	0.1924	1.660	0.01696	0.000293	0.556
4	3.09	0.01900	0.001529	6.70	0.550	0.532	1.916	0.1916	1.643	0.0362	0.000293	0.534
					0.546	0.528	1.890	0.1890	1.632	0.0357	0.000289	0.546
5	3.06	0.0384	0.001525	6.72	0.537	0.523	1.880	0.1880	1.600	0.0723	0.000287	0.504
					0.530	0.516	1.830	0.1830	1.579	0.0703	0.000279	0.530
6	3.00	0.0745	0.001515	6.81	0.504	0.497	1.745	0.1745	1.494	0.1299	0.000264	0.496
					0.502	0.496	1.733	0.1733	1.489	0.1291	0.000263	0.502

~ 1-041 ТН,0 — 160 У2.І.0 — 0 THf. O — 0

“Calculated from Eq. (4.89).

6 Calculated from Eq. (4.85).

“Calculated from Eq. (4.86).

4 Calculated from Eq. (4.87).

“Calculated from Eq. (4.88).

^Specified concentrations.

Stage no. m	Organic concentration, mol/liter Ут = Уі + ~xo)	Assumed total N03 ~ in aqueous phase s *N03~,m	Distribution coefficients		Aqueous concentration, mol/liter
	S _ Ут m D	Total N03 " concentration in aqueous phase" VS *N03-,m
HN03 T’H. m	Zr(N03)4 yZr, m	Free Hf(N03)4 TBP" s s T’Hf. m ^TBP. m
HNOa oHb	Zr(N03 )4 Dzrc	Hf(N03)4 DHfd	HN03 VS xH, m	Zr(N03)4 x Zr, m	Hf(N03)4 ^Hf. m
і	1.576r	0.0578Z	0.00000578-^ 0.558	6.57	0.532	1.859	0.1859	2.96	0.0311	31.1 X 10"*	6.58
				6.58	0.533	1.872	0.1872	2.96	0.0309	30.9 X 10"*	6.58
2	1.553	0.0727	0.0000206 0.549	6.60	0.526	1.837	0.1837	2.95	0.0396	0.0001122	6.61
				6.61	0.526	1.845	0.1845	2.95	0.0394	0.0001118	6.61
3	1.553	0.0768	0.0000594 0.543	6.64	0.523	1.834	0.1834	2.97	0.0418	0.000324	6.64
4	1.562	0.0780	0.0001613 0.532	6.68	0.516	1.809	0.1809	3.03	0.0431	0.000891	6.70
				6.70	0.517	1.822	0.1822	3.02	0.0428	0.000885	6.70
5	1.586	0.0784	0.000431 0.506

= 3.0

"Calculated from Eq. (4.88). * Calculated from Eq. (4.85). "Calculated from Eq. (4.86). d Calculated from Eq. (4.87). "Calculated from Eq. (4.89). ■^Specified concentrations.

The next step is to apply the material-balance equation (4.52) to calculate the concentrations in the organic stream leaving stage 2. The calculation then proceeds down through the scrubbing section by a repetition of these steps.

The large variation in distribution coefficients is noteworthy. This is caused principally by changes in the concentration of uncombined TBP. This large variation makes necessary the use of a numerical calculation method.

Figure 4.19 is a plot of zirconium concentration versus hafnium concentration in the organic phase, with points for the extracting section from Table 4.9 and points for the scrubbing section from Table 4.10. The point of intersection occurs at

Extracting: 5 < N < 6

Scrubbing; 4 < (M + 1) < 5

By interpolation, the point of intersection of the curves of Fig. 4.18 occurs at Extracting: N = 5.2

Scrubbing: M + 1 = 4.6

Thus, six theoretical stages in the extracting section and four in the scrubbing section would result in higher values of zirconium recovery and hafnium decontamination than those specified.

Finally, a check must be made to determine if organic concentrations of nitric acid in the extracting and scrubbing sections match at the feed point. That this condition is satisfied can be seen from Fig. 4.20, a plot of organic concentrations leaving a stage versus stage number. If nitric acid does not match at the feed point, a new value of nitric acid concentration in the organic extract must be assumed and the entire calculational procedure repeated.

Figure 4.20 and the McCabe-Thiele diagrams of Fig. 4.21 illustrate the principle of separation of two metals by solvent extraction with complexing agents. The largest change in organic concentration of the more extractable component, zirconium, occurs in the extracting section. After the first extracting stage the hafnium soon reaches a concentration in the organic that is almost in equilibrium with the hafnium in the aqueous feed entering the succeeding stages. The hafnium concentration in organic is actually reduced somewhat as the feed stage is

Figure 4.19 Zirconium and haf­nium concentrations in organic phase, to determine feed point in zirconium-hafnium separation example.

0. 5

Figure 4.20 Concentrations in organic phase leaving a stage as a function of stage number for zirconium-hafnium separation.

approached, because the decreasing concentration of uncombined TBP reduces the hafnium distribution coefficient. In the scrubbing section the lower flow ratio of aqueous to organic allows a relatively large decrease in the concentration of hafnium in the organic. Because some zirconium is removed from the organic in the scrubbing section, the maximum zirconium concentration occurs near the feed point, resulting in the lowest concentration of uncombined TBP at this point. The equilibrium lines in the extracting and scrubbing sections are essentially identical for zirconium, but not for hafnium.

The feed conditions have been chosen so that nitric acid in the organic feed is nearly in equilibrium with the nitric acid in the aqueous feed and scrub solution, so that little change in nitric acid concentration occurs through the cascade. The small changes in nitric acid concentration are due principally to variations in amount of TBP complexed by zirconium.

The SEPHIS [G6, H2, Wl] and SOLVEX [SI] computer codes are applications of the above techniques for calculating the number of equilibrium stages for the TBP extraction of uranium and plutonium in nitric acid solution. These codes include correlations of the

Figure 4.21 Stage concentration diagram for zirconium-hafnium extracting-scrubbing example.

extraction equilibrium constants and activity coefficients as affected by total ionic strength. Corrections for incomplete ionization of aqueous plutonium may also be important.